# Complex Polynomial of nth degree

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1. Apr 26, 2015

### Nathew

1. The problem statement, all variables and given/known data
Show that if
$$P(z)=a_0+a_1z+\cdots+a_nz^n$$
is a polynomial of degree $n$ where $n\geq1$ then there exists some positive number $R$ such that
$$|P(z)|>\frac{|a_n||z|^n}{2}$$
for each value of $z$ such that $|z|>R$

2. Relevant equations
Not sure.

3. The attempt at a solution
I've tried dividing through by the nth power of z. That way I can somehow incorporate the R value somehow but I'm not exactly sure where to go from here.

Thanks!

2. Apr 26, 2015

### PeroK

Maybe you could start by showing that for large enough z,

$|z|^n > |a_0|$

And, perhaps, rewrite the equation with everything but $a_n z^n$ on the LHS.

Can you see, without doing any algebra, why it's true?

Last edited: Apr 26, 2015